Advertisement

Stopped Compound Poisson Process and Related Distributions

  • Claude Lefèvre
Chapter
  • 2.3k Downloads
Part of the Statistics for Industry and Technology book series (SIT)

Abstract

This chapter considers the first-crossing problem of a compound Poisson process with positive integer-valued jumps in a nondecreasing lower boundary. The cases where the boundary is a given linear function, a standard renewal process, or an arbitrary deterministic function are successively examined. Our interest is focused on the exact distribution of the first-crossing level (or time) of the compound Poisson process. It is shown that, in all cases, this law has a simple remarkable form which relies on an underlying polynomial structure. The impact of a raise of a lower deterministic boundary is also discussed.

Keywords and phrases

Compound Poisson process first-crossing lower boundary ballot theorem generalized Abel-Gontcharoff polynomials generalized Poisson distribution quasi-binomial distribution damage model 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bhaskara Rao, M., and Shanbhag, D. N. (1982). Damage models, In Encyclopedia of Statistical Sciences (Eds., S. Kotz and N.L. Johnson), 2, pp. 262–265, John Wiley_& Sons, New York.Google Scholar
  2. 2.
    Böhm, W., and Mohanty, S. G. (1997). On the Karlin-McGregor theorem and applications, The Annals of Applied Probability, 7, 314–325.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Consul, P. C. (1974). A simple urn model dependent upon predetermined strategy, Sankhyã Series B, 36, 391–399.zbMATHMathSciNetGoogle Scholar
  4. 4.
    Consul, P. C. (1989). Generalized Poisson Distributions: Properties and Applications, Marcel Dekker, New York.zbMATHGoogle Scholar
  5. 5.
    Consul, P. C., and Mittal, S. P. (1975). A new urn model with predetermined strategy. Biometrische Zeitung, 17, 67–75.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Daniels, H. E. (1963). The Poisson process with a curved absorbing boundary, Bulletin of the International Statistical Institute, 40, 994–1008.MathSciNetGoogle Scholar
  7. 7.
    De Vylder, F. E. (1999). Numerical finite-time ruin probabilities by the Picard-Lefèvre formula, Scandinavian Actuarial Journal, 2, 97–105.CrossRefGoogle Scholar
  8. 8.
    Durbin, J. (1971). Boundary-crossing probabilities for the Brownian motion and Poisson processes and techniques for computing the power of the Kolmogorov-Smirnov, Journal of Applied Probability, 8, 431–453.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gallot, S. F. L. (1993). Absorption and first-passage times for a compound Poisson process in a general upper boundary, Journal of Applied Probability, 30, 835–850.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Ignatov, Z. G., Kaishev, V. K., and Krachunov, R. S. (2001). An improved finite-time ruin probability formula and its Mathematica implementation, Insurance: Mathematics and Economics, 29, 375–386.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ignatov, Z. G., and Kaishev, V. K. (2004). A finite-time ruin probability formula for continuous claim severities, Journal of Applied Probability, 41, 570–578.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Johnson, N. L., Kotz, S., and Kemp, A. W. (1992). Univariate Discrete Distributions, 2nd ed., John Wiley_& Sons, New York.zbMATHGoogle Scholar
  13. 13.
    Kotz, S., and Balakrishnan, N. (1997). Advances in urn models during the past two decades, In Advances in Combinatorial Methods and Applications to Probability and Statistics (Ed., N. Balakrishnan), pp. 203–257, Birkhäuser, Boston.Google Scholar
  14. 14.
    Lefèvre, Cl., and Picard, Ph. (1990). A nonstandard family of polynomials and the final size distribution of Reed-Frost epidemic processes, Advances in Applied Probability, 22, 25–48.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lefèvre, Cl., and Picard, Ph. (1999). Abel-Gontchareoff pseudopolynomials and the exact final outcome of SIR epidemic models (III), Advances in Applied Probability, 31, 532–549.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Perry, D. (2000). Stopping problems for compound processes with applications to queues, Journal of Statistical Planning and Inference, 91, 65–75.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Perry, D., Stadje, W., and Zacks, S. (1999). Contributions to the theory of first-exit times of some compound processes in queueing theory, Queueing Systems, 33, 369–379.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Perry, D., Stadje, W., and Zacks, S. (2002). Hitting and ruin probabilities for compound Poisson processes and the cycle maximum of the M/G/1 queue, Communications in Statistics: Stochastic Models, 18, 553–564.zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Picard, Ph., and Lefèvre, Cl. (1994). On the first crossing of the surplus process with a given upper boundary Insurance: Mathematics and Economics, 14, 163–179.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Picard, Ph., and Lefèvre, Cl. (1996). First crossing of basic counting processes with lower non-linear boundaries: a unified approach through pseudopolynomials (I), Advances in Applied Probability, 28, 853–876.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Picard, Ph., and Lefèvre, Cl. (1997). The probability of ruin in finite time with discrete claim size distribution, Scandinavian Actuarial Journal, 1, 58–69.Google Scholar
  22. 22.
    Picard, Ph., and Lefèvre, Cl. (2003). On the first meeting or crossing of two independent trajectories for some counting processes, Stochastic Processes and their Applications, 104, 217–242.zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Pyke, R. (1959). The supremum and infimum of the Poisson process, Annals of Mathematical Statistics, 30, 568–576.MathSciNetGoogle Scholar
  24. 24.
    Stadje, W. (1993). Distributions of first exit times for empirical counting and Poisson processes with moving boundaries, Communications in Statistics: Stochastic Models 9, 91–103.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Stadje, W., and Zacks, S. (2003). Upper first-exit times of compound Poisson processes revisited, Probability in the Engineering and Informational Sciences, 17, 459–465.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Takács, L. (1967). Combinatorial Methods in the Theory of Stochastic Processes, John Wiley_& Sons, New York.zbMATHGoogle Scholar
  27. 27.
    Takács, L. (1989). Ballots, queues and random graphs, Journal of Applied Probability, 26, 103–112.zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Zacks, S. (1991). Distributions of stopping times for Poisson processes with linear boundaries, Communications in Statistics: Stochastic Models, 7, 233–242.zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Zacks, S. (1997). Distributions of first exit times for Poisson processes with lower and upper linear boundaries, In Advances in the Theory and Practice of Statistics: A Volume in Honor of Samuel Kotz (Eds. N. L. Johnson and N. Balakrishnan), pp. 339–350, John Wiley_& Sons, New York.Google Scholar
  30. 30.
    Zacks, S., Perry, D., Bshouty, D., and Bar-Lev, S. (1999). Distributions of stopping times for compound Poisson processes with positive jumps and linear boundaries, Communications in Statistics: Stochastic Models, 15, 89–101.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Boston 2006

Authors and Affiliations

  • Claude Lefèvre
    • 1
  1. 1.Université Libre de BruxellesBruxellesBelgium

Personalised recommendations